Question

If $G\left( {3, - 5,r} \right)$ is centroid of triangle ABC where $A\left( {7, - 8,1} \right),B\left( {p,q,5} \right)$ and $C\left( {q + 1,5p,0} \right)$ are vertices of a triangle then values of $p,q,r$ are respectively.
A. $6,5,4$
B. $ - 4,5,4$
C. $ - 3,4,3$
D. $ - 2,3,2$

Answer 1

Hint:
We will use the formula of centroid, that is, if \[\left( {{x_1},{y_1},{z_1}} \right),\left( {{x_2},{y_2},{z_2}} \right),\left( {{x_3},{y_3},{z_3}} \right)\] are the coordinates of vertices of a triangle, then the coordinates of centre are given by $\left( {\dfrac{{{x_1} + {x_2} + {x_3}}}{3},\dfrac{{{y_1} + {y_2} + {y_3}}}{3},\dfrac{{{z_1} + {z_2} + {z_3}}}{3}} \right)$. Substitute the given values of centroid and vertices of a triangle. Next, compare the coordinates to determine the value of $p, q, r$.

Complete step by step solution:
We are given that $G$ is the centroid of the triangle, whose vertices are $A,B$ and $C$.
A centroid is a point of intersection of three medians of a triangle.
Also, we know that if \[\left( {{x_1},{y_1},{z_1}} \right),\left( {{x_2},{y_2},{z_2}} \right),\left( {{x_3},{y_3},{z_3}} \right)\] are the coordinates of vertices of a triangle, then the coordinates of centre are given by $\left( {\dfrac{{{x_1} + {x_2} + {x_3}}}{3},\dfrac{{{y_1} + {y_2} + {y_3}}}{3},\dfrac{{{z_1} + {z_2} + {z_3}}}{3}} \right)$
Then, from the given conditions, we will have,
$
  \left( {3, - 5,r} \right) = \left( {\dfrac{{7 + p + q + 1}}{3},\dfrac{{ - 8 + q + 5p}}{3},\dfrac{{1 + 5 + 0}}{3}} \right) \\
   \Rightarrow \left( {3, - 5,r} \right) = \left( {\dfrac{{8 + p + q}}{3},\dfrac{{ - 8 + q + 5p}}{3},2} \right)
$
Now, we will compare the coordinates and form respective equations.
On comparing the $x$ coordinates, we will get,
$
  3 = \dfrac{{8 + p + q}}{3} \\
   \Rightarrow 9 = 8 + p + q
 $
$ \Rightarrow p + q = 1$ eqn. (1)
On comparing the $y$ coordinates, we will get,
$
   - 5 = \dfrac{{ - 8 + q + 5p}}{3} \\
   \Rightarrow - 15 = - 8 + q + 5p
$
\[ \Rightarrow 5p + q = - 7\] eqn. (2)
On comparing the $z$ coordinates, we will get,
\[r = 2\]
We will solve equations (1) and (2) to determine the values of $p$ and $q$.
Now, subtract equations (1) and (2) to eliminate the value of $q$ and then find the value of $p$
$
  p + q - 5p - q = 1 - \left( { - 7} \right) \\
   \Rightarrow - 4p = 8 \\
   \Rightarrow p = - 2
$
Substitute the value \[p = - 2\] in equation(1) to find the value of $q$
$
   - 2 + q = 1 \\
   \Rightarrow q = 3
 $
Therefore, the values of $p,q,r$ are \[ - 2, 3, 2\]

Hence, option D is the correct option.

Note:
If two coordinates $\left( {{x_1},{y_1},{z_1}} \right)$ is equal to $\left( {{x_2},{y_2},{z_2}} \right)$, then ${x_1} = {x_2}$, ${y_1} = {y_2}$ and ${z_1} = {z_2}$. A centroid is a point of intersection of three medians of a triangle, where the median is a line from the vertex that divides the opposite side in two equal parts.